Project: Find the relationship among the volumes of a right circular
cone, a hemisphere and a right circular cylinder of equal radii and
equal heights.
Answers
Given :-
♦ A right circular cone, a hemisphere and a right circular cylinder of equal radii and equal heights.
To find :-
♦ The relationship among the volumes of the three solids .
Solution :-
Let the radius of a right circular cone be r units
Let the radius of a hemi sphere be r units
Let the radius of a right circular cylinder be
r units
Since , The three solids are having same radii.
Let the height of a right circular cone be h units
= Radius of the cone = r units
Let the height of a hemi sphere be h units
Height of the hemisphere = Radius of the hemisphere = r units
Let the height of a right circular cylinder be
h units
= Radius of the cylinder = r units
Since, The three solids are having equal heights
The volume of a right circular cone
= (1/3)πr²h cubic units
= (1/3)πr²×r
= (1/3)πr³ cubic units ---------(1)
The volume of a hemisphere
= (2/3)πr³ cubic units ----------(2)
The volume of the right circular cylinder
= πr²h cubic units
= πr²×r
= πr³ cubic units -------------------(3)
Now,
The ratio of their volumes
= V of Cone : V of Hemisphere : V of Cylinder
= (1/3)πr³ : (2/3)πr³ : πr³
= (1/3) : (2/3) : 1
On multiplying each term by 3 then
= 1:2:3
Answer :-
Relation -1:-
♦ The ratio of the volumes of the cone, hemisphere and cylinder is 1:2:3
Relation -2:-
♦ The volume of hemisphere is 2/3rd of the volume of the cylinder and the volume of cone is 1/3rd of the volume of the cylinder .
Used formulae:-
★The volume of a right circular cone
The volume of a right circular cone = (1/3)πr²h cubic units
★The volume of a hemisphere
The volume of a hemisphere = (2/3)πr³ cubic units
★The volume of the right circular cylinder
The volume of the right circular cylinder = πr²h cubic units
- r = radius
- h = height
- π = 22/7
UNDERSTANDING CONCEPT :-
First use the formula of volume of cone
which is given as :-
volume of cone = V = 1/3hπr²
where r is radius and h is height, then formula of
volume of cylinder which is given as
Volume of cylinder = πr²h
where r is radius and h is height and then use the that the volume of hemisphere which is given as
Volume of hemisphere = (2/3) πr
where r is radius. Then put the values according to the question and find their ratios by putting their volumes in this
Volume of cone: Volume of cylinder: Volume of hemisphere
QUESTION :-
Find the relationship among the volumes of a right circular cone, a hemisphere and a right circular cylinder of equal radii and
equal heights.
GIVEN :-
relationship among the volumes of a right circular
cone
hemisphere and a right circular cylinder of equal radii and equal heights.
TO FIND :-
Relationship between volume among the volume of right circular cone, a hemisphere and a right circular cylinder = ?
SOLUTION :-
Let, radius (r) = height (h) = x unit
Volume of right circular cone = πr² h /3 = πx/3
The volume of the right circular cylinder = πr²h
= πx
thus, from eq (i) , (ii) , (iii) we get :-
the relationship between the volume are as
follow
{ volume of the right circular cone} { volume of
the hemisphere } [Volume of the right circular cylinder}
= πx/3 : 2πx/3
cancelling all the similar terms
= 1/3 : 2/3 : 1
multiplying by 3 throughout
1 : 2 : 3
Hence, Relationship between volumes
among the volume of right circular cone, a hemisphere and a right circular cylinder-1 : 2 : 3